A computational statistics and stochastic modeling approach to

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A computational statistics and stochastic
modeling approach to materials-by-design
Nicholas Zabaras
Materials Process Design and Control
Laboratory Sibley School of Mechanical and
Aerospace Engineering188 Frank H. T. Rhodes
Hall Cornell University Ithaca, NY
14853-3801 Email zabaras_at_cornell.edu URL
http//www.mae.cornell.edu/zabaras/
Materials Process Design and Control Laboratory
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INFORMATION FLOW ACROSS SCALES
Engineering
Information flow
Materials
Filtering and two way flow of statistical
information
Chemistry
Physics
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Length Scales ( )
A
1
10
2
10
4
10
6
10
9
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OVERVIEW OF STOCHASTIC FRAMEWORK

Adaptive spectral and support methods

Multiscale information theoretic material
heterogeneity models
Existing deterministic application software
Stochastic analysis framework


Geometric, boundary, material uncertainties
Information theoretic correlation kernels


Accurate risk and reliability estimates
Complete statistical response of outputs

Explicit uncertainty quantification
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UNCERTAINTY IN FINITE DEFORMATION PROBLEMS
Metal forming Forging
velocity Lubrication friction at die-workpiece
interface Intermediate material state variation
over a multistage sequence residual-stresses,
temperature, change in microstructure,
expansion/contraction of the workpiece Die shape
is it constant over repeated forgings ? Damage
evolution through processing stages Preform
shapes (tolerances)
Material heterogeneity
Composites fiber orientation, fiber spacing,
constitutive model Biomechanics material
properties, constitutive model, fibers in tissues
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UNCERTAINTY ANALYSIS USING SSFEM
F(?)
xn1(?)
X
Bn1(?)
B0
xn1(?)x(X,tn1, ?,)
Key features Total Lagrangian formulation
(assumed deterministic initial configuration) Spec
tral decomposition of the current configuration
leading to a stochastic deformation gradient
Materials Process Design and Control Laboratory
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EFFECT OF UNCERTAIN FIBER ORIENTATION
Aircraft nozzle flap composite material,
subjected to pressure on the free end
Orthotropic hyperelastic material model with
uncertain angle of orthotropy modeled using KL
expansion with exponential covariance
Two independent random variables with order 4 PCE
(Legendre Chaos)
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MODELING INITIAL CONFIGURATION UNCERTAINTY
xn1(?)x(XR,tn1, ?,)
F(?)
X(?)
xn1(?)
FR(?)
Bn1(?)
B0
XR
F(?)
BR
Introduce a deterministic reference configuration
BR which maps onto a stochastic initial
configuration by a stochastic reference
deformation gradient FR(?). The deformation
problem is then solved in this reference
configuration.
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Strain Localization due to Uncertainty in Initial
Configuration
Deterministic simulation- Uniform bar under
tension with effective plastic strain of 0.7 .
Power law constitutive model.
Initial configuration assumed to vary uniformly
between two extremes with strain maxima in
different regions in the stochastic simulation.
Plastic strain 0.7
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Strain Localization due to Uncertainty in Initial
Configuration
Stochastic simulation
Results plotted in mean deformed configuration
Plastic strain 0.7
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Strain Localization due to Uncertainty in Initial
Configuration
Stochastic simulation
Plastic strain 0.7
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UNCERTAINTY DUE TO MATERIAL HETEROGENEITY
State variable based power law model. State
variable Measure of deformation resistance-
mesoscale property Material heterogeneity in the
state variable assumed to be a second order
random process with an exponential covariance
kernel. Eigen decomposition of the kernel using
KLE.
Eigenvectors
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UNCERTAINTY DUE TO MATERIAL HETEROGENEITY
Dominant effect of material heterogeneity on
response statistics
Load vs Displacement
SD Load vs Displacement
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CORRELATION KERNELS
Model Based Correlation Kernels
Information Theoretic transfer based on Wavelets
Process based microstructure models
Raw Material
Manufacture Process
Correlation Kernels at various scales in a SSFEM
framework
Multiscale parameters of stochastic deformation
problem
Correlation in wavelet domain A multiscale
analysis
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INFORMATION THEORETIC FRAMEWORK
Correlation kernels based on intrascale mutual
information criterion
Correlation kernels at macro scale
Wavelet coefficients at macro scale
Wavelet coefficients at meso scale
KLE effective method to model material
heterogeneity using correlation kernels. From
phenomenology to explicit derivation of kernels
using multiscale information Information transfer
and filtering between scales based on maximum
entropy criterion and wavelet parameters.
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INFORMATION THEORY AND MULTISCALE MODELING?

• Field of mathematics founded by
• Shannon in 1948
• Revolutionized the outlook towards communication
  of information (rigorous mathematical standpoint)
  
• Information Theory used to link simulations at
  various scales in multiscale simulations
• Try to transfer as much information as possible
  about parameters of interest (displacements,
  stresses, strains etc)

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AN INFORMATION THEORETIC VIEWPOINT
Informational entropy of parameter ensembles
during upscaling reduces due to averaging out of
fine details present at micro scale
How much information is required at each scale
and what is the acceptable loss of information
during upscaling to answer performance related
questions at the macro scale ?
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COMMUNICATION AND MULTISCALE ANALYSIS?
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INFORMATION TRANSFER ACROSS CHANNELS
Received messages
Sent messages
symbols
Channel coding
Source decoding
Channel decoding
Source coding
source
receiver
channel
Compression
Error Correction
Decompression
Channel Capacity
Source Entropy
Rate vs Distortion
Capacity vs Efficiency
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AN INTERESTING ANALOGY
Wavelet Basis at lower scale
Wavelet Basis at higher scale
Information Upscaling Channel
Wavelet based coding of parameters
Information Theoretic upscaling of
wavelet coefficients
Decoding of wavelet parameters
macro scale
micro scale
Source information
Received information
Information lost here
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NEED FOR WAVELETS
Schematic of wavelet representation

• A very useful tool in areas where a multiscale
  analysis is important.
• Could be used as a tool to quantify information
  of physical parameters of interest.
• Very useful for such analyses because it is
  mathematically compact and consistent

Information across all scales
Fo
Micro scale
Q1Fo
F1
F2
Q2Fo
Frantziskonis, Deymier (2000,2003)
Q3Fo
Information Lost
Wavelets as a multiscale tool

• Compound Wavelet Matrix Method Independent
  simulations done at two different scales and
  solutions obtained mapped onto wavelet domain.
• Use the above-mentioned to bridge the scales
  between atomic and continuum, both spatially as
  well as temporally

Meso scale
Fn
?a,b wavelet coefficients at scale a and
spatial location b.
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HOMOGENIZATION IN WAVELET SPACES
Full Microstructure Information
Homogenized Properties at next scale
Wavelet Basis
Complete Homogenization
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Wavelet based Reduced Order study
Information lost when approximated to fourth scale
Decreasing resolution of microstructure using
Daubechies-1 wavelets. Choose a scale with
truncated wavelet basis functions so that only
parameters above that scale could be resolved.
Tradeoff
Choose level of analysis so that computational
time is significantly reduced (at lower
resolutions) while ensuring that information loss
of the omitted wavelets is tenable
Chosen wavelet basis elements
Completely averaged scale.
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Mutual Information Comparison across scales
Daubechies Family
Biorthogonal Family
Mutual Information The information that
parameters in a scale are able to convey about
parameters in another scale. A higher information
loss occurs when we try to reduce the
dimensionality of the solution when the physics
involves lower order scales. Hence a hierarchical
wavelet based method to be employed while
ensuring that information lost in the truncated
wavelet bases is minimized.
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INFORMATION AND WAVELET MEASURES
Renyi vs Shannon
Entropy Measures

• Renyis and Shannons Entropy have the
• same minima
• Renyis quadratic entropy is computationally
• very efficient and fast
• Mean square error criterion for training is a
• very special case of Renyis mutual
• information maximization criterion

(Shannon)
(Tsallis)
(Renyi)
(a Scale parameter,b space parameter, w
wavelet coefficients)
Wavelet Maps

• Map parameters at lower scale onto a wavelet
  basis
• Upscale these coefficients by maximizing mutual
  information between multiscale wavelet
  coefficients
• Obtain the macro scale information maximized
  parameters

Wavelet Families
Haar Daubechies Biorthogonal Morlet
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INFORMATION THEORETIC DOWNSCALING
Averaged velocity gradient
Variations across averaged values as seen from
micro scale
Constant velocity gradient applied at the macro
scale to the specimen
Micro scale parameters would be distributed
across this macro value. Hence a stochastic
simulation needed at the micro.
MAXENT (Jaynes) The entropy of variables must be
maximized over the parameter space to obtain
micro parameters subjected to macro averages
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MAXENT as an upscaling tool
Experimental Simulations when microstructure
approximated as PV tessellations using MC
analysis (Kumar et al, 1992)
Microstructure Reconstruction via MAXENT MAXENT
provides means to obtain the entire
microstructural variability of entities whose
average and certain moments are available at
higher scales (Sobczyk, 2003)
A deterministic simulation at higher scale is
equivalent to a stochastic simulation at lower
scales where the stochastic parameters are
obtained using MAXENT and higher scale parameters
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INFORMATION LEARNING
Linskers maximum Mutual Information Mutual
information between desired signal and output
signal should be maximized
Desired Macroscale entities
Information Potential
Information Force
Normalized Information Potential
Basis Microstructures
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Information Theoretic Learning
Information Learning
Used to reduce the computational time when the
parameters needs to be transferred continuously
at each time step. Train a neural network with
Information criterion, that is mutual information
between actual and nn based outputs is maximized
A convergence study of neural network based
single level upscaling process employing
information theoretic criterion
Information potential of one implies that the nn
based output can predict exactly the result of
upscaling process
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MICROSTRUCTURE BASED MODELS
Model chosen based on microstructure
Orientation distribution function model
Lineal analysis of microstructure photograph
Poly-phase material
Dendritic
Pure metal
Spatial Correlation Structure of Models are known
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IDEA BEHIND STOCHASTIC VMS
Subgrid solution obtained semi-analytically using
element-wise stochastic Fourier transform approach
Coarse element
Compatibility of stochastic subgrid solution with
respect to large scale solution uncertainty is
examined The subgrid scale approximate stochastic
solution is substituted to large scale weak form
yielding a stabilized formulation
Process captured by large scale computational grid

• Variational multiscale decomposition
  Stochastic solution considered as a sum of two
  scale components viz. large scale and subgrid
  scale
• Large scale can be captured by the computational
  grid
• Subgrid scale has to be modeled
• The approximate subgrid effects are considered
  in the large scale variational formulation to
  yield a stochastic finite element method with
  stabilized properties

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STOCHASTIC INPUT-OUTPUT MODELING

• Uncertainty in inputs and outputs modeled by
  considering them as stochastic processes

Representation techniques for a Space-time
stochastic process
STOCHASTIC INPUT REPRESENTATION

• Stochastic inputs modeled using Karhunen-Loeve
  expansion based on spectral decomposition of the
  covariance kernel

are independent random variables that
form a basis for the input probability space

• For output representation schemes, we assume
  that input can be represented in terms of a few
  independent random variables

GLOBAL OUTPUT REPRESENTATION

• The output is represented in the above
  Wiener-Askey generalized polynomial chaos
  expansion
• Specific polynomials are chosen based on the
  input joint probability density function.
  Examples are GaussianHermite, UniformLegendre
  and BetaJacobi.

LOCAL OUTPUT REPRESENTATION

• Stochastic output is represented as a piecewise
  polynomial on the input stochastic support space
• We use a finite element mesh on the support
  space that is refined towards regions of high
  input PDF importance based gridding approach

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MORE ON SUPPORT-SPACE METHODOLOGY

• We assume that the stochastic input has been
  represented in the KL expansion
• represents the stochastic input vector
• represents the joint PDF of
  inputs
• is
  called the stochastic support space characterized
  by positive input joint PDF
• Any stochastic output can be represented as a
  function defined on this input space
• We consider a finite element piecewise
  representation

Each nodal point on the spatial mesh is linked to
a support-space grid
Statistics of relevance or complete PDF passed on
to the spatial mesh
Spatial domain with finite element grid
Grid on support-space
Importance spaced grid
Support-space of input
Two-level grid approach
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BIG PICTURE STOCHASTIC VMS FRAMEWORK
Localized subgrid problem solved using
FEM/wavelets in regions where multiscale physics
is important. GPCE for smooth solutions,
support-space for discontinuous solutions
Subgrid mesh
Using feedback control and posteriori error
estimates separate domain into multiscale and
non-multiscale regions
Multiscale physics important
Semi analytical solutions to reduced operator
problems dictate the subgrid boundary conditions
Multiscale physics not important
Renyi Shannon entropy, Linskers information
maximization theorem
Information theoretic filters based on entropy
criterion to ensure that only the relevant
statistical information is transferred from
subgrid to large scales
Modify large scale residual based on subgrid
solutions
Fully resolved solution
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EXPLICIT SUBGRID MODELING TECHNIQUES
Case with high subgrid stochasticity
Macro-residual passed to subgrid
Fully stochastic VMS decomposition

• Macro component discretized using finite element
  coarse mesh
• Each coarse element further contains a link to
  the subgrid mesh
• This linkage is null if the coarse element does
  not possess any multiscale behavior

For GPCE this means, the subgrid basis has higher
dimensional Wiener-Askey polynomials
For support-space, the subgrid support space is
represented in terms of hierarchical
support-space basis functions
Subgrid BCs calculated
Macro domain
Combining concept different subgrid equation
Explicit subgrid model
Volume-averaging concept valid far from mold and
at larger times
Generate subgrid basis functions forces by
partition of unity or wavelets
Melt
Near wall regions using higher-order transport
equations derived from ab-initio principles
Mold
Compatibility of boundary conditions, statistical
description and linking hypothesis developed in a
consistent mathematical framework Ability to
capture discontinuities in stochastic response,
mixed-random variables with both discrete and
continuous behavior
Modify large scale weak formGPCE for smooth
subgrid variations, support-space for
discontinuous subgrid solutions
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MULTISCALE REGION IDENTIFICATION

• As shown in figure, all points in macro
  component do not possess multiscale
  characteristic
• Multiscale identification toolbox consists of
  following algorithms with implementation
• Stochastic dual-problem based posterior error
  estimates
• Feedback control of posterior error
• Adaptive tree-cased algorithms for distribution
  of multiscale elements among different processors
• Mathematical techniques for parallel solution
  reconstruction

Multiscale identification toolbox based on
feedback control and posterior error estimates
Reconstructed solution after parallel computations

Feedback/ adaptive toolbox
Multiscale and transition regions distributed
separately among parallel processors for subgrid
calculations
Mathematical posterior error estimates
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FRAMEWORK APPLIED TO MICROSTRUCTURES
Large scale information captured
Variational consistent upscaling



3c
3b
3a
Meso-to-macro upscaling
Fine scale detail filtered using entropy based
information filter, updation of wavelet
coefficients
Wavelet homogenized properties, state variable
solutions passed from microstructure to subgrid
Macro-state variable statistics obtained by
upscaling the subgrid statistics using a
consistent VMS approach
Spectral stochastic FEM, RFB, Greens function
2a
2b
2c
Spectral stochastic FEM at macro level
Macro-to-meso downscaling
Statistics of large scale solutions for
macro-state variables represented in wavelet
expansions
Evolution of subgrid statistics driven by macro
residuals
Evolution of microstructure driven by
micromechanical models
1a
1b
1c
Component at start of processing stage
Approximate regions with multiscale physics
identified
Subgrid element mapped to microstructure clouds
Subgrid scale model equations
Three-scale VMS

• Three scale stochastic variational multiscale
  framework
• Upscaling and downscaling of statistical
  information is based on information theoretic
  concepts
• Wavelets are used instead of a finite element
  representation for the support-space output
  representation
• Stochastic homogenization applied to upscale
  from microstructure to subgrid scale

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STATISTICAL LEARNING TOOLBOX
Training samples
NUMERICAL SIMULATION OF MATERIAL RESPONSE
Update data In the library

• Multi-length
• scale analysis
• Polycrystalline
• plasticity

STATISTICAL LEARNING TOOLBOX
Image

• Functions
• Classification methods
• Identify new classes

ODF
Associate data with a class update classes
Process controller
Pole figures
Materials Process Design and Control Laboratory
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DYNAMIC MICROSTRUCTURE LIBRARY CONCEPTS
Space of all possible microstructures
A class of microstructures (eg. Equiaxial grains)
New class partition
Hierarchical sub-classes (eg. Medium grains)
Expandable class partitions (retraining)
distance measures
New class
Dynamic Representation
New microstructure added
Axis for representation
Updated representation
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Feature vector Three point probability function
Feature Autocorrelation function
3D Microstructures
3D Microstructures
Class - 1
g
r mm
Class - 2
LEVEL - 2
LEVEL - 1
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APPLICATIONS MICROSTRUCTURE RECONSTRUCTION
Process
Pattern recognition
Microstructure evolution models
2D Imaging techniques
Feature extraction
Reverse engineer process parameters
Database
vision
Microstructure Analysis (FEM/Bounding theory)
3D realizations
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ADAPTIVE REDUCED ORDER MULTI-STAGE PROCESS DESIGN
Initial Conditions- stage 2
Sensitivity of material property
Direct problem a
Sensitivity problem
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LIBRARY FOR TEXTURES
Uni-axial (z-axis) Compression Texture
110 fiber family
Feature
q fiber path corresponding to crystal direction
h and sample direction y
z-axis lt110gt fiber (BB)
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DATABASE DRIVEN COMPUTATIONAL PROCESS DESIGN
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EXAMPLE DESIGN FOR DESIRED MAGNETIC PROPERTY
Crystal lt100gt direction. Easy direction of
magnetization zero power loss
h
External magnetization direction
Stage 1 Shear 1 (a1 0.9745)
TWO STAGE PROCESS
Stage 2 Tension (a2 0.4821)
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Information transferred to subsequent length
scales used for statistical learning
Electronic scale First principle Atomistics
DFT, Monte Carlo simulations
Alloy systems Newly discovered compositions
Lattice energies, Interatomic potentials, Bulk
free energies, Interfacial free energies, Bulk
strength, crystal structure
Knowledge discovery
Electronic-scale database
Statistical learning
Information from other scales
To higher length scales
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Data from DFT
Statistical learning
Phase field model
Dislocation dynamics
Microstructure Morphology
Properties of individual phases and crystals
Model reduction
Microstructure Class Hierarchy
Autocorrelation
LEVEL - 1
Data-mining
Constitutive laws, Microstructure-dependent
properties through bounding theories and FEM
Expanded view of the meso-scale database
3 point probability
3D Microstructures
To stochastic continuum models
Meso-scale database
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ATOMISTIC STATISTICAL LEARNING
Property statistics Micro-alloying elements
Si
La
Bulk Modulus
Mn
Os
100 90 80
Atomic Al
Co
Ni
GPa per atomic
Zr
-1
0 1 D(Bulk Modulus) 3
Macro-property correlations with atomistic
properties
0 10 20
Atomic Li
Alloy property maps
Ductility
Pitting corrosion
Objective Increase Bulk Moduli descriptor
Desired property Ductility, corrosion resistance
Bulk Modulus
Bulk Modulus
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ATOMISTIC SCALE STATISTICAL LEARNING
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DESIGNING ALLOYS THROUGH STATISTICAL LEARNING
Meshing and virtual experimentation (OOF)
Diffusion coefficients
Phase field model
Thermodynamic variables (CALPHAD) Mobilities
Interfacial energies
Nucleation Models
Property statistics
Design problems 1) Determine the compositions
that give optimum properties
2) Design process sequences to obtain
desired properties
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Hyperplanes quantify correlation of local length
scale features with the objective and higher
length scale effects
DFT
Electron scale database
Alloy systems
Alloy Al-Ni Composition ?? (Atomistic) Process
variables ?? (Meso-Macro)
Phase Field
Statistical features at the local length scales
sYmax
Homogenized property (OOF)
Processes Microstructures (Phase field)
Desired strength distribution
Objective
Design decisions
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MATERIAL DESIGN THROUGH STATISTICAL LEARNING
Materials Process Design and Control Laboratory